# Questions tagged [groupoids]

A groupoid (in the sense of category theory) is a small category in which every morphism is an isomorphism. Groupoids arise throughout mathematics, e.g. in guise of fundamental groupoids in the theory of covering spaces, holonomy groupoids in the theory of foliations or Lie groupoids in mathematical physics. For groupoids in the sense of universal algebra, i.e., a set with a binary operation, please use the (magma) tag.

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### Exemples of applications of “groupoidification” to linear algebra

I just read Baez's very nice blog notes about groupoidification, and around the beginning, he states : "From all this, you should begin to vaguely see that starting from any sort of incidence ...
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### Is the 2-Category of Groupoids Locally Presentable?

I am wondering if the 2-Category of groupoids is Locally Presentable? Locally presentable means the category is accessible and co-complete. Edit: It has been pointed out that the category of ...
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### Why are categories not called monoids (and why are monoids not called… mons?) [closed]

If groupoids are "indexed groups", wouldn't that same naming scheme imply that categories should be called "monoidoids", or more sensibly, why aren't categories called "monoids" and monoids called... "...
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### What's the correct notion of equivalence in a double category?

What's the appropriate notion of equivalence between two objects in a double category? At first I thought the answer was just an equivalence in one of the associated $2$-categories, but then I ...
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### Equivalence of Lie groupoids $\phi: H \rightarrow G$ induces an equivalence of categories $\phi^*: G\text{-spaces} \rightarrow H\text{-spaces}$.

In Orbifolds as Groupoids there is the notion of an equivalence $\phi: H \rightarrow G$ between Lie groupoids (2.4) and of $G$-spaces (5.1). Given a smooth functor $\phi: H \rightarrow G$ we can ...
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### Equivalence between the Dwyer-Kan loop groupoid and the fundamental groupoid

Let $X$ be a homotopy 1-type (a space with vanishing homotopy groups above degree one). It is a classical fact that $X$ can be recovered completely from its fundamental groupoid. On the the other ...
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### Inversion on an internal groupoid

Let $\mathcal{C}$ be a category with pullbacks, with $\mathscr{G}=({\bf Ob}_\mathscr{G},{\bf Hom}_\mathscr{G},cod,dom,{\bf 1}, \circ_{\mathscr{G}},-^{-1})$ an internal groupoid in $\mathcal{C}$. I'm ...
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### Faithful representation of $C_c(X \times X)$.

Let $X$ be a smooth manifold. $X \times X$ is product manifold. $\mu$ is a Borel measure on $X$. There are two aims (i) Associate a $C^*$ norm to $C_c(X\times X)$, making it a $C^*$ algebra. (ii)...
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### Groupoid Theory I, Higson's notes

This is Example 7.22, page 100. We first consider $G=TM$ a smooth groupoid with base space $M$. The source and range maps are the same projection maps $s,r:TM \rightarrow M, X_m \mapsto m$. Now I ...
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### Milnor construction and deloopings

To construct a classifying space (and universal bundle) of a topological group $G$ one can use the well-known Milnor construction based on the infinite join of $G$. On the other hand one can (at ...
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### What is a topological groupoid?

I'm reading section 2.7 (Fundamental Group of the Circle) of the book Algebraic Topology by Tom Dieck. The section mentions the term "topological groupoid" but I cannot find the definition in previous ...
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### Is the 2-category of groupoids a topos?

I have no justification for this, but I am wondering if the 2-category of groupoids is a topos.
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### Space of Riemannian metrics as a topological groupoid

I'm reading these notes on groupoids and I'm struggling with example 1.4. I recall the relevant definitions below. Definition: A groupoid $\mathcal{G}$ is a small category in which every arrow is ...
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### Fibered Categories in Groupoids

I'm reading an article of Aaron Mazel-Gee about Fibered categories in grupoids and there is an example which I don't understand. Here the full article: https://etale.site/writing/stax-seminar-talk.pdf ...
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### Can nonisomorphic groupoids have homotopy equivalent classifying spaces?

We know that two discrete groups having the same classifying space up to homotopy are isomorphic. One can just take fundamental groups and conclude. The situation with topological groups is subtler. ...
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### Right (bi)adjoint of the inclusion of $\mathbf{Grpd}$ in $\mathbf{Cat}$

Let $\mathbf{Grpd}$ and $\mathbf{Cat}$ be respectively the 2-categories of small groupoids and of small categories. At the 1-categorical level, the inclusion $\mathbf{Grpd}\rightarrow\mathbf{Cat}$ has ...
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### what is an ∞-group?

I was reading on nLab and I found the term infinity group. The definition is awfully abstract: An ∞-group is a group object in ∞Grpd. Equivalently (by the delooping hypothesis) it is a pointed ...
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### Is every topological groupoid equivalent to a disjoint union of topological groups?

It's a fact that any groupoid is equivalent to a disjoint union of (deloopings of) groups. See, e.g. Proposition 4.3 of https://ncatlab.org/nlab/show/groupoid#PropertiesEquivalencesOfGroupoids. Does ...
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### The Whitehead product and $\pi_{\leq 3} S^2$

Why does "the non-vanishing of the Whitehead bracket" imply that the fundamental 3-groupoid $\pi_{\leq 3} S^2$ of the two-shere cannot be strictified (as claimed here)?
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### The $\mathbb{C}$ vector space monad on the 2-Category of Groupoids

In this post, I am asking about the existence of something called the "Vector Space Monad" on the 2-Category of groupoids (Grpd). In a comment, it was pointed out that the monad should exist due to ...
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### The étale topos of a scheme is the classifying topos of…?

By a theorem of Joyal and Tierney, every Grothendieck topos is the classifying topos of a localic groupoid. It has been proved (e.g. C. Butz. and I. Moerdijk. Representing topoi by topological ...
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### Are there examples of unital and nuclear $C^*$-algebras satisfying the UCT that are not groupoid algebras of an amenable etale groupoid?

Jean Louis Tu showed that the (maximal) groupoid $C^*$-algebra of a groupoid satisfying the Haagerup property (which includes all amenable groupoids) will satisfy the UCT. I am curious if there are ...
Let $G$ be a (finite) group, containing a subset $H$. We suppose that $H$ contains the identity and that it is closed under taking inverses. What are some of the algebraic properties of subgroups ...